Evolving Linguistic Fuzzy Models from Data Streamsdanfl7/Evolving_linguistic... · Evolving Linguistic Fuzzy Models from Data Streams 3 2 Fuzzy Set Based Evolving Modeling FBeM is

Evolving Linguistic Fuzzy Models fromData Streams

Daniel Leite and Fernando Gomide

Abstract This work outlines a new approach for online learning from imprecisedata, namely, fuzzy set based evolving modeling (FBeM) approach. FBeM is anadaptive modeling framework that uses fuzzy granular objects to enclose uncer-tainty in the data. The FBeM algorithm is data flow driven and supports learningon an instance-per-instance recursive basis by developing and refining fuzzy mod-els on-demand. Structurally, FBeM models combine Mamdani and functional fuzzysystems to output granular and singular approximations of nonstationary functions.In general, approximand functions can be time series, decision boundaries betweenclasses, and control and regression functions. Linguistic description of the behaviorof the system over time is provided by information granules and associated rules.An application example on a reactive control problem, underlining the complemen-tarity of Mamdani and functional parts of the model, illustrates the usefulness ofthe approach. More specifically, the problem concerns sensor-based robot naviga-tion and localization. In addition to precise singular output values, granular outputvalues provide effective robust obstacle avoidance navigation.

1 Introduction

Evolving models of systems from data stream is a common and challenging taskin many fields such as machine learning, data mining, signal processing, computervision, and others. About a decade ago, processing data flows in online environ-ment motivated evolving intelligent systems (EIS) [1]-[8], a modeling frameworkweaponed with flexible tools to support dynamics and analysis of complex systems.

Recent research on EIS has considered granular systems [9] for learning fromsequential observations or data streams [7]-[8]. The granularity of information ex-plicitly embedded into granular systems offers key features in dynamic modelingsuch as transparency and flexibility of models.

Daniel Leite and Fernando GomideUniversity of Campinas, School of Electrical and Computer Engineering, Sao Paulo, Brazile-mail: [email protected]; [email protected]

1

2 Daniel Leite and Fernando Gomide

A formal granular system [10] relies on the concepts of information granule, quo-tient structure and granular mapping to encapsulate uncertainty contained in datastreams, and turn information into knowledge. A granular mapping is defined overinformation granules and a quotient structure. Mapping granules consists in associ-ating a set of granules expressed in some input space to another set of granules drawin an output space. Granular mappings are frequently encountered in rule-based sys-tems, where the mapping is given by If-Then type of statements. Computing withgranules emphasizes multiple levels of understanding, analyzing and representinginformation. Zadeh first introduced the notion of information granulation in 1979[11] and pointed fuzzy set theory as a potential foundation.

Starting with imprecise description of the values of attributes, namely, measure-ments affected by noise and disturbances, we can represent them in terms of a formalfuzzy object into the realm of fuzzy granular computing. The purpose is represent-ing the meaning of an attribute using conceptual entities (information granules andassociated rules) and making no prior assumption about the statistical properties ofthe data. The linguistic appealing fuzzy representation ‘z is like γ’ would convey theessence of the values of the attributes.

Mamdani [12] and functional [13] rule-based systems are largely known typesof fuzzy systems which emerged about thirty years ago from studies in linguisticsynthesis and control. Both systems share common antecedent and differ from eachother by employing fuzzy set-based or functional consequents. Mamdani modelsare known to be more linguistically interpretable than functional models whereasfunctional models are generally more precise. Interpretability and accuracy requiretradeoffs and one usually prevails over the other. This work combines Mamdani andfunctional models within a single granular modeling FBeM approach. FBeM takesadvantage of the granular and singular responses given by both models simultane-ously.

FBeM models fall in the category of granular computing models because theyuse fuzzy-type information granules to construct granular mappings, and associateinput granular datum to output granular datum. Fuzzy granules trade the generalityof the structure of the data and give algorithms with simple math and rules describ-ing their meaning. Fundamentally, FBeM system adapts its structure when a newconcept appears in a data stream. By structure we mean fuzzy information gran-ules, If-Then rules and a concept. Granulation eases recursive structural updatingand maintenance of synopsis of data with modest storage and processing require-ments. Experts usually prefer that online systems give approximated outputs as wellas tolerance bounds of the approximations. Building fuzzy sets from imprecise mea-surements is the FBeM approach for housing noise and disturbances.

The remaining of this work is organized as follows. Section 2 addresses the gran-ular fuzzy modeling approach from data streams. Section 3 presents the necessarydefinitions and notation. Section 4 details the FBeM learning algorithm. Section 5considers an application example in which the FBeM system operates as an evolv-ing controller for robust autonomous navigation. Section 6 concludes the work andsuggests issues for further investigation.

Evolving Linguistic Fuzzy Models from Data Streams 3

2 Fuzzy Set Based Evolving Modeling

FBeM is a data-driven evolving modeling approach that aims at providing simul-taneous singular and granular function approximation and linguistic description ofthe behavior of a system. An FBeM-based model conveys a set of If-Then rules ex-tracted from data streams. The collection of rules means a granular representationof a complex system. Learning in FBeM creates and develops local models, namely,information granules and rules, using single-scan-through-the-data recursive algo-rithm and streaming data instances. The resulting granular mapping, union of localmodels, is a global model of the system.

FBeM rules manage information granules. For each information granule thereexists a corresponding rule and a linguistic description. In general, no granules andrules need necessarily exist prior learning. They are gradually evolved over time.Experts may wish to provide a verbal description about the system using their intu-ition and experience. Evolving fuzzy modeling, Fig. 1, supports both, learning fromdata flows and learning from experience.

Fig. 1 Evolving fuzzy modeling

In FBeM models, rules Ri governing information granules γ i are of the type

Ri: IF (x1 is Ai1) AND ... AND (x j is Ai

j) AND ... AND (xn is Ain)

THEN (y1 is Bi1) AND y1 = pi

1(x j∀ j) AND...(yk is Bi

k) AND yk = pik(x j∀ j) AND

...(ym is Bi

m)︸︷︷︸Mamdani

AND ym = pim(x j∀ j)︸︷︷︸

functional

,

where x j and yk are variables of the data stream (x,y)[h], h = 1, ...; Aij and Bi

k aremembership functions built in light of the data being available; pi

k are approxima-tion polynomials. The collection of rules Ri, i = 1, ...,c, casts a rule base. Rules inFBeM are created and adapted on-demand whenever the structure of the data callsfor improvement in the current model. It is worth noting that an FBeM rule combines


both, linguistic and functional consequents. The linguistic part of the consequent ismore linguistically appealing once a fuzzy set may come with a label. The func-tional part of the consequent offers precision. Using the proposed structure, FBeMtakes advantage of both models within a single modeling framework.

FBeM systems grant important characteristics for adaptive modeling. Continu-ous online processing on an instance-per-instance recursive basis enables FBeM todeal with nonstationary behavior, that is, concept drift. Dealing with nonstationarydata stream includes detecting and tracking changes in the time-space structure ofthe underlying data. The FBeM approach for data flow mining and knowledge dis-covery relies predominantly on constructive bottom-up modeling procedures, butallows decomposition-based top-down procedures. In the next sections we discussadaptive mechanisms to develop antecedent and consequent parameters of FBeMrules. Details about structural learning in online environment are addressed subse-quently in Section 4.

2.1 Rule Antecedent

Granulation of attributes A j, j = 1, ...,n, and Bk, k = 1, ...,m, within FBeM modelsis based on scatter fuzzy partitioning. Scatter partitioning uses fuzzy sets Ai

j and Bik,

refinements of A j and Bk, which can be extended to fuzzy hyperboxes in a productspace by means of alpha level sets. The scattering process clusters the data into in-formation granules γ i when appropriate (low-level granules represent more specificconcepts and facilitates comprehension when supported by a context) and consid-ers the coexistence of different granularities in the data. Granules are positioned atarbitrary locations into the product space. An aspect to be taken into account withscattering-type granulation refers to searching for a suitable amount of partitions,their positions and sizes. Figure 2 illustrates the scatter granulation mechanism.

Fig. 2 Scattering type of granulation


Learning in unknown environment claims for opportune creation of fuzzy struc-tural objects. Fitting data into conveniently placed and sized granules through scat-tering leaves substantial flexibility for incremental recursive modeling. The FBeMapproach particularly seizes fuzzy hyperboxes as formal granular objects to encloseuncertainty in the data of streams. This concedes considerable freedom in choosingthe internal structure of the granular object. FBeM admits online adaptation proce-dures applied over trapezoidal fuzzy objects.

Yager [14] has demonstrated that a trapezoidal fuzzy subset Aij = (li

j,λij,Λ

ij,L

ij)

allows the modeling of a wide class of granular objects. Triangular, interval andpoint valued subsets are special cases of trapezoids. An interval fuzzy subset isa trapezoid Ai

j where lij = λ i

j and Λ ij = Li

j. A triangular fuzzy subset refers to atrapezoid where λ i

j = Λ ij. A singleton is a trapezoid where li

j = λ ij = Λ i

j = Lij.

Additional features that make trapezoidal representation attractive is the ease ofacquiring the necessary parameters. Four parameters related to real features need tobe captured. They are not cognitively complex features and come straight from adata stream. Many operations on trapezoids can be performed using the endpointsof intervals, level sets of trapezoids. The linearity of the trapezoidal representa-tion allows calculation of only two level sets to obtain a complete implementation.Trapezoids can be translated quite easily to linguistic propositions.

2.2 Rule Consequent

Consequent of FBeM rules joins Mamdani and functional fuzzy models to approxi-mate actual system outputs and provide tolerance bounds on the approximation. TheMamdani, linguistic part of the consequent depicts information granules Bi

k occur-ring along the domain of output variables k whereas the functional part pi

k comprisescorresponding singular local functions.

Similarly to the approach for embodying antecedent of rules Aij, consequents of

rules Bik benefit from scattering-based granulation and fuzzy hyperboxes to cluster

output data streams skillfully. We assume trapezoidal fuzzy subsets Bik = (ui

j, υ ij, ϒ i

j ,U i

j) to assemble granular objects in the output space by the same motivations previ-ously described for Ai

j. Local functions pik hold for all instances measured from the

actual function f and that rest inside the fuzzy hyperbox delineated by the granuleγ i. In general, each pi

k can be of different type and is not required to be linear. TheFBeM system suggested in this work adopts affine local functions of the type

pik = ai

0k +n

∑j=1

aijkx j

for simplicity. Higher order polynomials can be used to approximate f . However, thenumber of coefficients to be estimated in this case increases substantially, especiallywhen the number or attributes n is large. FBeM approximands pi

k aim to estimate


f within the domain of information granule γ i. The recursive least mean square(RLMS) algorithm can be used to determine the coefficients ai

jk of pik.

Trapezoidal representation and scatter partitioning allow granules to overlap.Therefore, two or more granules can accommodate a data instance. FBeM singu-lar output is determined as a weighted mean value over all rules,

pk =

c∑

i=1min(Ai

1, ...,Ain)pi

k

c∑

i=1min(Ai

1, ...,Ain)

as a way to deal with regional conflicts and ensure smooth transition between piecesof superposed local functions.

Granular output given by [uij,U

ij] may enrich decision making and sometimes

represents more useful information than the more specific numerical output pk.Whilst being very specific using pk we risky being incorrect, being unspecific from[ui

j,Uij] we turn ourselves assure of being correct. However, sacrificing accuracy

pays the price of the guarantee of correctness. Information granules tend to reflectthe essence of the structure of the underlying data stream and emphasize the inter-pretability of the result.

3 Definitions and Notation

Consider a trapezoidal membership function Aij = (li

j,λij,Λ

ij,L

ij) as illustrated in

Fig. 3. Aij is a granular object, a fuzzy subset, used to model imprecise data over

x j. Similarly, let Bik = (ui

k,υik,ϒ

ik ,U

ik) be a granular object to model imprecise data

over yk. Subsets Ai = (Ai1, ...,A

ij, ...,A

in) and Bi = (Bi

1, ...,Bik, ...,B

im) assemble an

information granule γ i governed by a rule Ri. For each rule antecedent Ai there is adirect correspondent consequent Bi.

Fig. 3 Trapezoidal membership function


We denote the support and the core of a trapezoidal membership function Aij as

supp(Aij) = [li

j,Lij] and

core(Aij) = [λ i

j,Λij],

respectively.The width of a membership function Ai

j is the lenght of its support, namely

wdt(Aij) = Li

j− lij.

Assume that ρ j and σk are the maximum width that membership functions Aij

and Bik of a granule may take in the input and output spaces, respectively. Values of

ρ and σ dictate the granularity (coarser, finer) of information granules and controlthe shape of membership functions. Suitable choices of ρ and σ are very importantbecause they impact model accuracy. A mechanism to capture the granularity of thedata adaptively according with the pace of the changes in a data stream is addressedlater in Section 4.2.

The midpoint of a subset Aij is the average of the bounds of its core, that is,

mp(Aij) =

λ ij +Λ i

j

2.

The union and intersection of trapezoidal subsets Aij over the j-axis is defined by

the maximum and minimum values of their individual membership functions:

A1j ∪ ...∪Ac

j = max(A1j , ...,A

cj),

A1j ∩ ...∩Ac

j = min(A1j , ...,A

cj).

The convex hull of Aij, namely ch(A1

j , ...,Acj), is a trapezoidal fuzzy subset that en-

closes all the elements of Aij independently of their intersection. Then:

ch(A1j , ...,A

cj) = (min(l1

j , ..., lcj),min(λ 1

j , ...,λcj ), ...

... max(Λ 1j , ...,Λ

cj ),max(L1

j , ...,Lcj)).

It follows that A1j ∪ ...∪Ac

j ⊆ ch(A1j , ...,A

cj) for any trapezoidal subsets Ai

j. Convexhull operation holds for intersecting and non-intersecting subsets.

Specificity measures refer to the amount of information conveyed by a fuzzysubset [15]. We use specificity to characterize the amount of information containedin FBeM estimations. Values of specificity range within the [0,1] interval. The valueapproaches 1 as the representative membership function closes in a single element.


Yager [14] defines the specificity of a trapezoidal membership function Aij as

sp(Aij) = 1−

wdt(Aij(0.5))

wdt(ch(A1j , ...,A

cj))

.

This simply means one minus the width of the 0.5 level set of Aij divided by the

width of the convex hull of all existing trapezoids along the j axis. In terms of theparameters of the membership functions we get

sp(Aij) = 1− 1

2

((Λ i

j +Lij)− (λ i

j + lij)

max(L1j , ...,L

cj)−min(l1

j , ..., lcj)

).

The specificity of an information granule γ i is

sp(γ i) = min(sp(Ai1), ...,sp(Ai

n),sp(Bi1), ...,sp(Bi

m)).

The concept of information specificity is highly correlated with the concept of in-formation granularity. Specificity measurements give an idea about the tightness ofexisting granules and how meaningful the rules managing the granules are.

4 Recursive Online Learning

FBeM learns online from a sequence of instances (x,y)[h], h = 1, ..., where y[h] isknown given x[h] or will become known some steps latter. Each pair (x,y) is anobservation of the target function f . When f changes with the time we say that thefunction is nonstationary. Modeling nonstationary functions requires tracking time-varying functions f [h]. Learning from sequential observations consists in executinga learning procedure capable of deciding when and how to perform structural andparametric adaptation of models based on measurements of f .

The learning procedure to evolve granular systems FBeM can be summarized asfollows:

BeginDo

1: Input a new instance (x,y)[h], h = 1, ...2: Accommodate possible new information

2.1: Create a new information granule and a rule2.2: Adapt some existing granules and rules

3: Discard instance (x,y)[h], h = 1, ...4: Refine the granular mapping

End


Steps 1 and 3 of the learning procedure stress the essence of data stream based adap-tive algorithms where instances are read and discarded one by one. Historical datais dispensable and evolution stands continuously. Granular systems evolve when-ever new information appears in the data, step 2. When a new instance does not fitcurrent knowledge, the procedure creates a new information granule and a rule gov-erning the granule, step 2.1. Conversely, if a new instance fits current knowledge,the procedure adapts existing information granules and rules, step 2.2. Eventually,the granular mapping may be optimized and refined, step 4. A number of very largescale problems can only be handled considering evolving approaches that, such asFBeM, do not scale with the number of streaming instances, and processing is con-strained by single pass over the data, memory and time [1]-[8]. Next sections detailthe FBeM learning procedure.

4.1 Creating rules

In FBeM no rule necessarily exists before learning starts. Rules are created andevolved as data are input. A new granule γc+1 is created adding a rule Rc+1 to thecurrent collection of rules R = {R1, ...,Ri, ...,Rc}. A rule is created either when atleast one input variable, say x[h]j , does not fit Ai

j ∀i, or y[h]k does not fit Bik ∀i. Fuzzy

connective operators based on t-norms suggest that both Aij∀ j and Bi

k∀k suit (x,y)[h]

for the corresponding rule to be considered. By contrary, a new rule is created toaccommodate the new never-seen-before information.

A new information granule γc+1 has input membership functions Ac+1j whose pa-

rameters are lc+1j = λ

c+1j =Λ

c+1j = Lc+1

j = x[h]j ∀ j, and output membership functions

Bc+1k with parameters uc+1

k = υc+1k =ϒ

c+1k =Uc+1

k = y[h]k ∀k. Thus, the new granulehas a singleton representation with full specificity. Initially, the coefficients of pc+1

k

are set as ac+1jk = 0, j 6= 0, and ac+1

0k = y[h]k ∀k.

4.2 Adapting rules

Adaptation of existing rules Ri either expands or contracts the support and the coreof rules antecedent Ai

j and consequent Bik to accommodate new data, and simultane-

ously adjusts the coefficients of local approximation functions pik.

A rule Ri may be adapted whenever an instance (x,y)[h] rests into the region ofgranule γ i. This means, geometrically, that the instance lay inside the fuzzy hyper-box of γ i or close enough so the granule is allowed to expand to include (x,y)[h].Referring to the trapezoidal membership function illustrated in Fig. 3, six situationsmay happen depending on where the instance is confined. They are:


If x[h]j ∈ [mp(Aij)−

ρ j2 , l

ij] then li

j(new) = x[h]j (support expansion)

If x[h]j ∈ [lij,λ

ij] then λ i

j(new) = x[h]j (core expansion)

If x[h]j ∈ [λ ij,mp(Ai

j)] then λ ij(new) = x[h]j (core contraction)

If x[h]j ∈ [mp(Aij),Λ

ij] then Λ i

j(new) = x[h]j (core contraction)

If x[h]j ∈ [Λ ij,L

ij] then Λ i

j(new) = x[h]j (core expansion)

If x[h]j ∈ [Lij,mp(Ai

j)+ρ j2 ] then Li

j(new) = x[h]j (support expansion)

Operations on the core parameters λ ij and Λ i

j require further adjustment of the mid-point of the granule:

mp(Aij)(new) =

λ ij(new)+Λ i

j(new)

2.

As result, support contraction may happen in two occasions:

If mp(Aij)(new)− ρ j

2 > lij then li

j(new) = mp(Aij)(new)− ρ j

2If mp(Ai

j)(new)+ ρ j2 < Li

j then Lij(new) = mp(Ai

j)(new)− ρ j2 .

Adaptation of the fuzzy sets of rules consequents Bik uses data y[h]k . Polynomial

coefficients aijk can be updated using the standard RLMS algorithm and taking ad-

vantage of the new instance that has activated γ i and its corresponding rule. Storageof a number of recent instances may be useful to guide alternative coefficient iden-tification algorithms, e.g., data chunks oriented algorithms. However, it comes withsome additional cost concerning memory and processing time.

A simple procedure we use in FBeM to adjust the maximum width, ρ and σ , ofgranules over time is as follows. Let β be the number of rules created after a certainnumber of evolution steps H. If the number of rules grows faster than a thresholdrate value η , then ρ and σ are increased by a factor (1+(β -η)/H) during the nextsteps. Otherwise, if the number of rules grows at a rate smaller than η , then ρ and σ

are decreased by the same factor (1+(β -η)/H). This procedure is useful to deal withdata stream granularity and let ρ and σ learn values for themselves. Initial values ofρ and σ are defined over non-normalized data.

Trapezoidal membership functions and scattering-type granulation allow gran-ules to overlap. Conflict resolution helps to choose which FBeM rule to adapt givena streaming instance. Conflict resolution is needed when two or more granules ac-commodate current data. An approach for conflict resolution is to select the granulewith the largest specificity to be adapted, that is,

i? = arg maxi(sp(γ i)).

Granule γ i? provides the tightest envelope for the input, and therefore a more concisedescription of the model behavior.


4.3 Refining the granular mapping

Procedures to refine the granular mapping include combination of neighbor gran-ules, covering gaps and deleting rules. Refinements are done after a certain numberof processing steps and contribute to develop smoother approximands and keep in-formation updated.

Combination of two neighbor granules, say γ1 and γ2, into a unique granuleformed by their convex hull, γψ = ch(γ1,γ2), is justified whenever the neighborsplace close enough to each other so that the specificity of the resulting granule isgreater than a threshold ε , i.e., sp(γψ)≥ ε . Merging granules reduces the number ofrules in the rule base and helps to eliminate partially overlapping granules conveyingsimilar information.

A situation converse to that of merging granules arises when the convex hull oper-ation ch(γ1,γ2) produces an information granule γψ whose size is too large accord-ing to the threshold value ε . Alternatively, when sp(γψ)< ε , gaps can be filled eval-uating the endpoints of nonoverlapping neighbors. The approach we use here is togenerate a new granule γc+1 in which Ac+1

j = (L1j ,L

1j , l

2j , l

2j ), Bc+1

k = (U1k ,U

1k ,u

2k ,u

2k),

that is, interval fuzzy subsets, ac+1jk = 0, j 6= 0, and ac+1

0k = (U1k + u2

k)/2. Depend-ing on the relative localization of the granules we must order the lower and upperbounds of the resulting membership functions accordingly. The approach to fill thegaps with granules is simple and particularly useful to extend the current model andto reduce data order dependency.

Nonstationarity may cause rules revision. Rules can be removed from the rulebase when they become inactive during a certain number of evolution steps. Thismay mean that the concept changed and deletion of granules is justified to keep therule base size compact and to conserve approximation efficiency.

5 Application example

5.1 Sensor-based robust navigation

We consider an instance of autonomous robot navigation in unknown environmentwith obstacles avoidance. From the control point of view, autonomous navigationproblem consists in designing driving rules based on available sensor data. TheFBeM system for sensor-based navigation plays the role of a reactive evolving con-troller that prevents the robot from colliding with obstacles. We assume that the pairof sensors available for obstacle detection are infrared sensors directed head-on,symmetrically, as shown in Fig. 4. Measurements from sensors SL and SR give alinear approximation of the surface of an obstacle. The control variable is the wheelsteering angle φ . Variable θ stands for the reference angle between the robot andthe border of the track.


Fig. 4 Environment for sensor-based navigation

We assume the navigation environment is flat, but unknown. Coordinates z1 andz2 range between [0,3000] and [0,5000]. Positive values of the steering angle φ rep-resent clockwise rotation of the steering wheel, and negative values mean counter-clockwise rotation. At every processing step, the controller outputs a steering angle.Input sensor readings, SL and SR, are proportional to the distance between robot andobstacle and limited to 500. The perpendicular distance between infrared beams is40. We want the robot to drive through the path without hitting the borderline.

Simple kinematical relations approximate the robot movement. For example, ifthe robot moves from position (z1,z2) to position (z′1,z′2) at step h with speed S, then

θ ′ = θ +φ

z′1 = z1 +S sin(θ ′)z′2 = z2 +S cos(θ ′)

Obstacle avoidance models often ignore physical limitations and processing delays.Estimated paths are often unrealistic once the feasibility of the trajectory is notguaranteed. In addition, uncertainty in measurements may hinder the robot to fol-low trajectories precisely. FBeM systems deal with these constraints by keeping therobot between tolerance bounds [u,U ] around the more precise estimation p.

Experiments concerning different navigation speeds and noise in the data wereconducted. Experts provided a few common-sense associations of how the state and


control variables behave prior to learning and navigation. Three rules were initiallyprovided, they are:

R1 : IF (SL is big) AND (SR is big) THEN (φ is zero) AND (φ = p11(SL,SR))

R2 : IF (SL is small)AND(SR is big)THEN(φ is positive)AND(φ = p21(SL,SR))

R3 : IF (SL is big)AND(SR is small)THEN(φ is negative)AND(φ = p31(SL,SR))

where the parameters of functions pi1 are a1

1 = (0,−0.034,0.034), a21 = (5,0.04,0.1),

and a31 = (−5,−0.1,−0.04). Trapezoidal membership functions which define the

subsets ‘big’, ‘small’; ‘negative’, ‘zero’ and ‘positive’ are shown in Fig. 5.

Fig. 5 Initial fuzzy membership functions

The robot is chosen to be initially placed at position (1900,100) with referenceangle θ = 0o in all experiments, as illustrated in Fig. 4. The following parametervalues are set over non-normalized data to evaluate the FBeM behavior: ρ [0] = 250,σ [0] = 80, η = 2, H = 100 and ε = 0.2. The online learning procedure is kept oncontinuously. It is worth noting that although the support of the initial membershipfunctions of Fig. 5 covers the whole variables domain, searching for more specificrules to fit never-seen-before streaming data may contract granules and thereforeactivate structural adaptation of models.

Figure 6 shows different trajectories concerning the robot driving at speeds 5, 10,20 and 30. We run FBeM algorithm four times independently in this experiment. Wenotice in the figure that the robot responded faster to obstacles detection driving atlower speeds. Moreover, alignment (parallel to the obstacle) after left and right turnstended to be more accurate at lower navigation speeds. Alignment yields smootherand shorter paths, which are intuitively preferable. Some classes of problems em-phasize fast environment exploration though. The figure on the right evidences thenumerical output p provided by the functional part of the FBeM controller, and thegranular output [u,U ] given by the bounds of the Mamdani part of FBeM conse-quent. Granular output is interpreted as a guaranteed safe path for navigation.

The simulation concerning speed 30, see Fig. 6, started with 3 rules and endedup with 4 rules. After contraction and drifting of initial membership functions ofantecedents toward frequently requested regions around 500, when the robot ap-proached the obstacle and sensor readings get smaller quickly, a new rule:

R4 : IF (SL is very small) AND (SR is small)THEN (φ is big positive) AND (φ = p4

1(SL,SR))

was created to help the robot turn right abruptly.


Fig. 6 FBeM navigation at different speeds

Experiment adding noise in the range ϑ = [−0.05,0.05] to the input data wasconducted. Noise may swing the robot from one side to the other. In this experimentthe robot speed remains fixed at 5 during the simulations. FBeM initial parametervalues are the same as in the previous experiment. Figure 7 shows the trajectoriesfrom independent simulations. We notice that when obstacles are out of vision, thecontroller accepts input data as they are and lets the robot explore the environmentfreely. Otherwise, when obstacles are detected, the controller responds turning therobot left and right satisfactorily. Smoothness on granules transitions alleviates un-likely swing effect.

Fig. 7 FBeM navigating with noisy input

Naturally, using more sensors and considering speed as control variable can im-prove the accuracy of FBeM systems for navigation. FBeM offers model-free es-timation of the control system. Even if a mathematical model is available, FBeMcontrollers may prove more robust, easier to adapt, and give additional linguisti-cally interpretable granular information, which may help design and analysis. Ifexperts can provide structured knowledge of the control system or if training dataare unavailable, the FBeM approach proceeds as an adaptive controller.


6 Conclusion

This work has suggested fuzzy set based evolving modeling as a framework to learnfrom online data streams. The FBeM algorithm recursively granulates data instancesto output singular and linguistic granular approximations of nonstationary functions.We addressed autonomous navigation problem in unknown environments as an ap-plication example. The FBeM system was able to evolve a reactive controller forrobust obstacle avoidance. The system combines good accuracy of functional fuzzymodels with the advantage of better semantic interpretation of linguistic models.Further work shall discuss different forms of manifestation of information granulesin data streams and the role FBeM approach to capture the essence of the informa-tion in the data.

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